Paralogistic {actuar} | R Documentation |

Density function, distribution function, quantile function, random generation,
raw moments and limited moments for the Paralogistic distribution with
parameters `shape`

and `scale`

.

dparalogis(x, shape, rate = 1, scale = 1/rate, log = FALSE) pparalogis(q, shape, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE) qparalogis(p, shape, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE) rparalogis(n, shape, rate = 1, scale = 1/rate) mparalogis(order, shape, rate = 1, scale = 1/rate) levparalogis(limit, shape, rate = 1, scale = 1/rate, order = 1)

`x, q` |
vector of quantiles. |

`p` |
vector of probabilities. |

`n` |
number of observations. If |

`shape, scale` |
parameters. Must be strictly positive. |

`rate` |
an alternative way to specify the scale. |

`log, log.p` |
logical; if |

`lower.tail` |
logical; if |

`order` |
order of the moment. |

`limit` |
limit of the loss variable. |

The paralogistic distribution with parameters `shape`

*= a* and `scale`

*= s* has density:

*
f(x) = a^2 (x/s)^a / (x [1 + (x/s)^a]^(a + 1))*

for *x > 0*, *a > 0* and *b > 0*.

The *k*th raw moment of the random variable *X* is
*E[X^k]*, *-shape < k <
shape^2*.

The *k*th limited moment at some limit *d* is *E[min(X, d)^k]*, *k > -shape*
and *shape - k/shape* not a negative integer.

`dparalogis`

gives the density,
`pparalogis`

gives the distribution function,
`qparalogis`

gives the quantile function,
`rparalogis`

generates random deviates,
`mparalogis`

gives the *k*th raw moment, and
`levparalogis`

gives the *k*th moment of the limited loss
variable.

Invalid arguments will result in return value `NaN`

, with a warning.

`levparalogis`

computes the limited expected value using
`betaint`

.

See Kleiber and Kotz (2003) for alternative names and parametrizations.

The `"distributions"`

package vignette provides the
interrelations between the continuous size distributions in
actuar and the complete formulas underlying the above functions.

Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon

Kleiber, C. and Kotz, S. (2003), *Statistical Size Distributions
in Economics and Actuarial Sciences*, Wiley.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012),
*Loss Models, From Data to Decisions, Fourth Edition*, Wiley.

exp(dparalogis(2, 3, 4, log = TRUE)) p <- (1:10)/10 pparalogis(qparalogis(p, 2, 3), 2, 3) ## variance mparalogis(2, 2, 3) - mparalogis(1, 2, 3)^2 ## case with shape - order/shape > 0 levparalogis(10, 2, 3, order = 2) ## case with shape - order/shape < 0 levparalogis(10, 1.25, 3, order = 2)

[Package *actuar* version 3.1-4 Index]